Having won a border deal with Mexico, the Trump administration turned its attention to a slow-moving Congress on Tuesday, demanding that lawmakers immediately approve a $4.5 billion emergency assistance package to care for the children and families surging at the border.

Senate Republicans said they will speed a bill through committee next week and dared Democrats to deny the border crisis and vote against the legislation.

President Trump requested the money six weeks ago, but it has been stuck in a legislative stalemate as Democrats first disputed the nature of the border crisis and then balked at the scope of the president’s requests.

Some daylight poked through Tuesday as Senate Minority Leader Charles E. Schumer, New York Democrat, said there is a chance for a deal — though the president won’t get all the money he wants.

“The humanitarian aid is a sweet spot,” he said, referring to $3.3 billion in money slated to go to the Health and Human Services Department, which is about to run out of money to house and care for unaccompanied migrant children.

He wasn’t so sure about the rest of the money.

Acting Homeland Security Secretary Kevin McAleenan said the entire package is essential and suggested that even more may be needed.

“The border is at its breaking point,” he said.

He has been sounding an alarm for six months, saying the deplorable conditions all sides have now seen at the border — overcrowding, children sleeping underneath a bridge, agents spending as much as half their time babysitting migrants instead of on patrol — are partly the result of lack of money.

But he also said Congress needs to close the loopholes that draw the migrants to the U.S. Foremost among those is a 2015 court ruling by a federal district judge in California who set a 20-day limit on holding illegal immigrant families.

Cases take 40 to 50 days to complete, so the 20-day limit means the families are released into communities. In one recent test period, 90% of them failed to show up for their court hearings once they were released.

Mr. McAleenan, who was in Guatemala late last month, said that message has been heeded. He recalled a conversation with one Guatemalan man whose daughter had left for the U.S. with her child.

“He said everybody knows that a child is a passport to migration to the United States,” the acting secretary said.

Sen. Lindsey Graham, South Carolina Republican and chairman of the Judiciary Committee, has written legislation to close that incentive. His bill would allow families to be held, together, for up to 100 days, giving enough time to complete most of the cases.

It would also create an asylum “timeout.”

Central Americans would be restricted from making asylum claims at the U.S. border but would be allowed to apply from outside the U.S., giving those with valid cases the chance to make them without having to make the dangerous trek.

“If you change that, I think 90% of this would stop,” Mr. Graham said.

The hiccup is that Democrats reject Mr. Graham’s and Mr. McAleenan’s reasoning.

Senate Minority Whip Richard J. Durbin of Illinois, the Democrats’ point man on immigration, said part of the problem lies in rough conditions in Central America. He says the answer is nation-building to produce a better society.

He said the other problem is Mr. Trump himself. Mr. Durbin said the travel ban, attempts to end the Obama-era Deferred Action for Childhood Arrivals deportation amnesty, moves to limit temporary protected status for hundreds of thousands of migrants, and last year’s zero-tolerance border policy that led to family separations have all backfired and left Democrats wary of any Trump moves.

“Make no mistake: Donald Trump takes pride in being the most anti-immigrant president in modern history, and at the same time his border security strategy is producing the worst results in our history,” Mr. Durbin said.

An average of 4,000 unauthorized migrants are encountered at the U.S. border each day, and nearly three-quarters of those are unaccompanied children or families.

Mr. McAleenan said the number of fraudulent families — in which adults falsely claim to be parents of the children they are bringing — is up to 4,800 so far this year.

“We’ve uncovered multiple child smuggling rings where innocent children are being used multiple times by different adults to gain illegal entry and release in the United States,” he said.

He pointed to the case of a 51-year-old man who paid $80 “to rent a 6-month-old child” to try to fake a family at the border.

Mr. Durbin said those cases shouldn’t be tolerated but shouldn’t sour the welcome for other illegal immigrant families.

The disagreements over causes and solutions could play out in the border spending bill.

Mr. Durbin said if Republicans are going to push for humanitarian assistance for children snared at the border, then they should also debate bills granting citizenship rights to millions of illegal immigrants already in the U.S.

Other Democrats have signaled that they want restrictions attached to the money limiting cooperation between HHS officials and the Homeland Security Department.

The Trump administration has said it wants to be able to share information across the government, such as instances in which the process of scouting out a sponsor to take an illegal immigrant child reveals a top deportation target living in the home.